
are equal and independent of \(i\)  for \(2 \le i \le d-1\) .
For the rest of this paragraph, assume that \(d \ge 3\) .
Let \(\beta +1\)  denote the common value of (REF ).
The scalar \(\beta \)  is called the fundamental constant of \(A,A^*\) .
For a parameter array of \(A,A^*\)  the entries
satisfy numerous relations [1]}.
In [2]}
the solutions are given in closed form,
in terms of seven free variables in addition to \(d\)  and \(\beta \) .
These seven variables are called basic.
The closed forms depend on the nature of \(\beta \) , as we now describe.
The Leonard pair \(A,A^*\)  is said to have type I whenever \(\beta \ne \pm 2\) ;
type II whenever \(\beta = 2\)  and \(\text{\rm Char}(\mathbb {F}) \ne 2\) ;
type III\(^+\)  whenever \(\beta =-2\) , \(\text{\rm Char}(\mathbb {F}) \ne 2\) , and \(d\)  is even;
type III\(^-\)  whenever \(\beta =-2\) , \(\text{\rm Char}(\mathbb {F}) \ne 2\) , and \(d\)  is odd;
type IV whenever \(\beta =2\)  and \(\text{\rm Char}(\mathbb {F})=2\) .
For each type, the solutions are given in [2]}.
